Constrained Colouring and σ-Hypergraphs

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Abstract

A constrained colouring or, more specifically, an (α, β)-colouring of a hypergraph H, is an assignment of colours to its vertices such that no edge of H contains less than α or more than β vertices with different colours. This notion, introduced by Bujtás and Tuza, generalises both classical hypergraph colourings and more general Voloshin colourings of hypergraphs. In fact, for r-uniform hypergraphs, classical colourings correspond to (2, r)-colourings while an important instance of Voloshin colourings of r-uniform hypergraphs gives (2, r −1)-colourings. One intriguing aspect of all these colourings, not present in classical colourings, is that H can have gaps in its (α, β)-spectrum, that is, for k1 < k2 < k3, H would be (α, β)-colourable using k1 and using k3 colours, but not using k2 colours.

In an earlier paper, the first two authors introduced, for being a partition of r, a very versatile type of r-uniform hypergraph which they called -hypergraphs. They showed that, by simple manipulation of the param- eters of a σ -hypergraph H, one can obtain families of hypergraphs which have (2, r − 1)-colourings exhibiting various interesting chromatic proper- ties. They also showed that, if the smallest part of is at least 2, then H will never have a gap in its (2, r − 1)-spectrum but, quite surprisingly, they found examples where gaps re-appear when α = β = 2.

In this paper we extend many of the results of the first two authors to more general (α, β)-colourings, and we study the phenomenon of the disappearance and re-appearance of gaps and show that it is not just the behaviour of a particular example but we place it within the context of a more general study of constrained colourings of σ -hypergraphs.

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Discussiones Mathematicae Graph Theory

The Journal of University of Zielona Góra

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CiteScore 2017: 0.64

SCImago Journal Rank (SJR) 2017: 0.633
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Target Group

researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs

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